Library / Tracking and estimation / TRK-06
Tracking and estimation
RMSE from bias and spread
Root-mean-square error of an estimator whose errors have a constant bias b and a random spread σ.
- Ref
- TRK-06
- Rev
- v1
- Field
- Tracking and estimation
- Result
- RMSE [m]
If an estimator's errors have a constant bias b and a random spread σ (a standard deviation), the mean-square error is b² + σ², so RMSE = √(b² + σ²). Bias and spread add in quadrature: a small bias barely matters beside a large spread, and the reverse. The RMSE should be evaluated against an independent reference, with the timing aligned.
Root-mean-square error of an estimator whose errors have a constant bias b and a random spread σ. The bias and the spread add in quadrature, so a bias can dominate even when the scatter looks small.
Calculator
- Mean-square error, b² + σ²13 m²
- Share of the mean-square error due to bias30.77 %
- RMSE with the bias removed, σ3 m
- RMSE with the spread removed, |b|2 m
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- With a bias of 2 m and a spread of 3 m the RMSE is 3.606 m. The bias supplies 30.8 % of the mean-square error, and removing it would leave an RMSE of 3 m.
- Quadrature addition means a bias of a third of the spread raises the RMSE by only about 5 %, but a bias equal to the spread raises it by 41 %. Whether to chase bias or noise depends on which is the larger.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (m) | Local step | Change in result |
|---|---|---|---|---|---|---|
| σ Standard deviation of the error | 0.5 to 10 m | 2.062 | 10.2 | 8.136 | +10% | +7.02 % |
| b Mean error (bias) | 0 to 10 m | 3 | 10.44 | 7.44 | +10% | +3.18 % |
Sweep of Standard deviation of the error
| Standard deviation of the error (m) | Root-mean-square error (m) |
|---|---|
| 0.5 | 2.062 |
| 1.688 | 2.617 |
| 2.875 | 3.502 |
| 4.063 | 4.528 |
| 5.25 | 5.618 |
| 6.438 | 6.741 |
| 7.625 | 7.883 |
| 8.813 | 9.037 |
| 10 | 10.2 |
Sweep one variable
Variables and how each one contributes
Root-mean-square error
What the equation gives you. Everything else feeds it.
Mean error (bias)
The mean error, in m. It is the average of the errors, which may be positive or negative; only its size matters here. Adds in quadrature, so it matters little when it is small compared with the spread. Raising it by 10 % (to 2.2) moves the result by +3.18 %, from 3.606 to 3.72 m.
Standard deviation of the error
The standard deviation of the errors about their mean, in m. Adds in quadrature with the bias. Spread is the larger part at the defaults. Raising it by 10 % (to 3.3) moves the result by +7.02 %, from 3.606 to 3.859 m.
Worked example with the values above
- Squares: b² = 4 and σ² = 9, summing to 13.
- Square root: √13.
- RMSE = 3.606 m.
Assumptions and limits
- Errors that have a constant bias and a fixed spread, evaluated against an independent reference with the timing aligned.
- Sufficient samples for the mean and standard deviation to be meaningful.
Common mistakes
- Evaluating against a reference that is itself noisy, or that is not aligned in time. Bias, timing misalignment and data dropouts then distort the RMSE.
- Quoting the RMSE as the average error. It weights large errors more strongly than the mean absolute error.
Statistics and uncertainty
The RMSE of a finite record is itself an estimate. For N independent Gaussian errors its relative uncertainty is about 1/√(2N): roughly 10 % for 50 samples and 3 % for 500. Errors that are correlated in time, as tracking errors usually are, carry less information per sample than that suggests.
Where it is used
- Scoring a tracker or navigation solution against truth.
- Deciding whether to reduce a bias (calibration) or the noise (filtering) first.
Maths you need
- Mean and standard deviation: bias and spread of a set of errors
- Quadrature: independent contributions add as squares